The Nationals held a 5-3 lead entering the ninth inning but Desmond's error at shortstop and a wild pitch by reliever Rafael Soriano allowed the Reds to force extra innings.
Desmond atoned for his miscue with a 439-foot shot into the upper deck off J.J. Hoover (0-2), and Wilson followed with a long drive over the center field bullpen -- his second homer of the game.
Craig Stammen (1-0) allowed an RBI double to Brandon Phillips in the bottom of the 11th, but then held on to record the victory.
Jayson Werth went 3-for-5 with a solo homer for Washington, which rebounded from a 15-0 pounding in Friday's series opener.
Todd Frazier provided three hits with an RBI and Shin-Shoo Choo hit a ninth-inning homer for Cincinnati.